---
tag: 47WJ34NC9K
kind: concept
name: Classical orthogonal polynomials
msc: [33C45, 42C05]
sources:
  - url: https://en.wikipedia.org/wiki/Classical_orthogonal_polynomials
same_as:
  wikidata: Q17006917
  wikipedia: Classical orthogonal polynomials
---

## Summary

The classical orthogonal polynomials are the three families of Jacobi, Laguerre and [[Hermite polynomials|MGACY98E9H]].
They include special cases such as the Gegenbauer, Legendre and [[Chebyshev polynomials|M6EAR1JHPP]].
Each family is orthogonal for a weight on an interval.
The weights are of beta type on $(-1, 1)$, of gamma type on $(0, \infty)$, and Gaussian on $\R$.
They are the only orthogonal polynomial families that are eigenfunctions of a second-order differential operator, and they have Rodrigues formulas and explicit recurrences.
They are the basic tools of approximation, Gaussian quadrature, and separation of variables in mathematical physics.

## Explanations

### L1 human

Two polynomials are orthogonal for a weight when the integral of their product times the weight is zero, like perpendicular vectors.
For a given weight, the Gram–Schmidt process applied to $1, x, x^2, \dots$ gives one orthogonal polynomial of each degree.
For three kinds of weight this produces families with especially nice formulas, and these are the classical ones.
For example, the weight $e^{-x^2}$ on the whole line gives the Hermite polynomials $1$, $2x$, $4x^2 - 2$, and so on.